WASHINGTON, April 7 (Reuters) - The White House on Monday warned Russian President Vladimir Putin against moving "overtly or covertly" into eastern Ukraine and said there was strong evidence that pro-Russian demonstrators in the region were being paid.

The Ukraine government in Kiev said the overnight seizure of public buildings in three cities in eastern Ukraine's mainly Russian-speaking industrial heartland were a replay of events in Crimea, the Black Sea peninsula Moscow seized and annexed last month.

"We saw groups of pro-Russian demonstrators take over government buildings in the eastern cities of Kharkiv, Donetsk and Luhansk," White House spokesman Jay Carney told reporters. "There is strong evidence suggesting some of these demonstrators were paid and were not local residents.

"If Russia moves into eastern Ukraine either overtly or covertly this would be a very serious escalation. We call on President Putin and his government to cease all efforts to destabilize Ukraine and we caution against further military intervention," Carney said.

Carney said the United States was concerned about Russia's moves and said Washington was studying whether to impose more sanctions.

"The president and his team will continue to assess Russia's actions and whether or not to impose those further sanctions," Carney said. "We also have the authorities to impose further sanctions for the transgressions already made by Russia when it comes to Crimea." (Reporting by Steve Holland and Jeff Mason; Editing by Sandra Maler)

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